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0. 前言
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0.1. 关于本书
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0.2. 如何使用本书
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0.3. 小结
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z" / > < / svg >
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1. 初识算法
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1. 初识算法
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1.1. 算法无处不在
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1.2. 算法是什么
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1.3. 小结
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z" / > < / svg >
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2. 复杂度
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2. 复杂度
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2.1. 算法效率评估
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2.2. 时间复杂度
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< a href = "../../chapter_computational_complexity/space_complexity/" class = "md-nav__link" >
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2.3. 空间复杂度
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2.4. 小结
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< a href = "../../chapter_data_structure/" class = "md-nav__link " >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z" / > < / svg >
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3. 数据结构
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3. 数据结构
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3.1. 数据结构分类
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3.2. 基本数据类型
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3.3. 数字编码 *
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< a href = "../../chapter_data_structure/character_encoding/" class = "md-nav__link" >
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3.4. 字符编码 *
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3.5. 小结
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4. 数组与链表
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4. 数组与链表
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4.1. 数组
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4.2. 链表
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< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/list/" class = "md-nav__link" >
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4.3. 列表
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4.4. 小结
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< div class = "md-nav__link md-nav__container" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z" / > < / svg >
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5. 栈与队列
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5. 栈与队列
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5.1. 栈
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< span class = "md-ellipsis" >
5.2. 队列
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
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5.3. 双向队列
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
5.4. 小结
< / span >
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< / a >
< / li >
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< / ul >
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< / li >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_hashing/" class = "md-nav__link " >
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< span class = "md-ellipsis" >
6. 散列表
< / span >
< / a >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_7_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_7" >
< span class = "md-nav__icon md-icon" > < / span >
6. 散列表
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_map/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.1. 哈希表
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_collision/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.2. 哈希冲突
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_algorithm/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.3. 哈希算法
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.4. 小结
< / span >
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< / a >
< / li >
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< / ul >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_tree/" class = "md-nav__link " >
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< span class = "md-ellipsis" >
7. 树
< / span >
< / a >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_8" >
< span class = "md-nav__icon md-icon" > < / span >
7. 树
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.1. 二叉树
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_tree_traversal/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.2. 二叉树遍历
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/array_representation_of_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.3. 二叉树数组表示
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_search_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.4. 二叉搜索树
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.5. AVL 树 *
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.6. 小结
< / span >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_9" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_heap/" class = "md-nav__link " >
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8. 堆
< / span >
< / a >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_9_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_9" >
< span class = "md-nav__icon md-icon" > < / span >
8. 堆
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/heap/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.1. 堆
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.2. 建堆操作
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/top_k/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.3. Top-K 问题
< / span >
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< / a >
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.4. 小结
< / span >
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< / a >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_10" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_graph/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z" / > < / svg >
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9. 图
< / span >
< / a >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_10_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_10" >
< span class = "md-nav__icon md-icon" > < / span >
9. 图
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.1. 图
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.2. 图基础操作
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.3. 图的遍历
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.4. 小结
< / span >
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< / a >
< / li >
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< / ul >
< / nav >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_searching/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z" / > < / svg >
< span class = "md-ellipsis" >
10. 搜索
< / span >
< / a >
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< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_11_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
10. 搜索
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.1. 二分查找
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search_edge/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.2. 二分查找边界
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.3. 哈希优化策略
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.4. 重识搜索算法
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.5. 小结
< / span >
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< / a >
< / li >
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< / ul >
< / nav >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_12" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_sorting/" class = "md-nav__link " >
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< span class = "md-ellipsis" >
11. 排序
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_12_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
11. 排序
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/sorting_algorithm/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.1. 排序算法
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/selection_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.2. 选择排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.3. 冒泡排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.4. 插入排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.5. 快速排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.6. 归并排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/heap_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.7. 堆排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/bucket_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.8. 桶排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/counting_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.9. 计数排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/radix_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.10. 基数排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.11. 小结
< / span >
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< / a >
< / li >
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< / ul >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_13" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_divide_and_conquer/" class = "md-nav__link " >
2023-07-19 16:10:17 +08:00
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< span class = "md-ellipsis" >
12. 分治
< / span >
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< span class = "md-status md-status--new" title = "最近添加" >
< / span >
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< / a >
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< label class = "md-nav__link " for = "__nav_13" >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_13_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_13" >
< span class = "md-nav__icon md-icon" > < / span >
12. 分治
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_divide_and_conquer/divide_and_conquer/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
12.1. 分治算法
< / span >
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_divide_and_conquer/binary_search_recur/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
12.2. 分治搜索策略
< / span >
< span class = "md-status md-status--new" title = "最近添加" >
< / span >
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< / a >
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< li class = "md-nav__item" >
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< a href = "../../chapter_divide_and_conquer/build_binary_tree_problem/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
12.3. 构建树问题
< / span >
< span class = "md-status md-status--new" title = "最近添加" >
< / span >
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< li class = "md-nav__item" >
< a href = "../../chapter_divide_and_conquer/hanota_problem/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
12.4. 汉诺塔问题
< / span >
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< / span >
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< / a >
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< li class = "md-nav__item" >
< a href = "../../chapter_divide_and_conquer/summary/" class = "md-nav__link" >
< span class = "md-ellipsis" >
12.5. 小结
< / span >
< span class = "md-status md-status--new" title = "最近添加" >
< / span >
< / a >
< / li >
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< / ul >
< / nav >
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< / li >
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13.3. 子集和问题
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13.4. N 皇后问题
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13.5. 小结
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14.1. 初探动态规划
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14.2. DP 问题特性
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14.2. DP 问题特性
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目录
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14.2.1. 最优子结构
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14.2.2. 无后效性
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14.3. DP 解题思路
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14.5. 完全背包问题
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14.7. 小结
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15. 贪心
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15.5. 小结
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16.1. 编程环境安装
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16.2. 一起参与创作
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参考文献
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参考文献
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目录
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14.2.1. 最优子结构
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14.2.2. 无后效性
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
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< h1 id = "142" > 14.2. 动态规划问题特性< a class = "headerlink" href = "#142" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 在上节中，我们学习了动态规划是如何通过子问题分解来求解问题的。实际上，子问题分解是一种通用的算法思路，在分治、动态规划、回溯中的侧重点不同：< / p >
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< ul >
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< li > 「分治算法」递归地将原问题划分为多个相互独立的子问题，直至最小子问题，并在回溯中合并子问题的解，最终得到原问题的解。< / li >
< li > 「动态规划」也对问题进行递归分解，但与分治算法的主要区别是，动态规划中的子问题是相互依赖的，在分解过程中会出现许多重叠子问题。< / li >
2023-07-26 03:16:04 +08:00
< li > 「回溯算法」在尝试和回退中穷举所有可能的解，并通过剪枝避免不必要的搜索分支。原问题的解由一系列决策步骤构成，我们可以将每个决策步骤之前的子序列看作为一个子问题。< / li >
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< p > 实际上，动态规划常用来求解最优化问题，它们不仅包含重叠子问题，还具有另外两大特性：最优子结构、无后效性。< / p >
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< h2 id = "1421" > 14.2.1. 最优子结构< a class = "headerlink" href = "#1421" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 我们对爬楼梯问题稍作改动，使之更加适合展示最优子结构概念。< / p >
< div class = "admonition question" >
< p class = "admonition-title" > 爬楼梯最小代价< / p >
< p > 给定一个楼梯，你每步可以上 < span class = "arithmatex" > \(1\)< / span > 阶或者 < span class = "arithmatex" > \(2\)< / span > 阶，每一阶楼梯上都贴有一个非负整数，表示你在该台阶所需要付出的代价。给定一个非负整数数组 < span class = "arithmatex" > \(cost\)< / span > ，其中 < span class = "arithmatex" > \(cost[i]\)< / span > 表示在第 < span class = "arithmatex" > \(i\)< / span > 个台阶需要付出的代价，< span class = "arithmatex" > \(cost[0]\)< / span > 为地面起始点。请计算最少需要付出多少代价才能到达顶部？< / p >
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< p > 如下图所示，若第 < span class = "arithmatex" > \(1\)< / span > , < span class = "arithmatex" > \(2\)< / span > , < span class = "arithmatex" > \(3\)< / span > 阶的代价分别为 < span class = "arithmatex" > \(1\)< / span > , < span class = "arithmatex" > \(10\)< / span > , < span class = "arithmatex" > \(1\)< / span > ，则从地面爬到第 < span class = "arithmatex" > \(3\)< / span > 阶的最小代价为 < span class = "arithmatex" > \(2\)< / span > 。< / p >
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< p > < img alt = "爬到第 3 阶的最小代价" src = "../dp_problem_features.assets/min_cost_cs_example.png" / > < / p >
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< p align = "center" > Fig. 爬到第 3 阶的最小代价 < / p >
< p > 设 < span class = "arithmatex" > \(dp[i]\)< / span > 为爬到第 < span class = "arithmatex" > \(i\)< / span > 阶累计付出的代价，由于第 < span class = "arithmatex" > \(i\)< / span > 阶只可能从 < span class = "arithmatex" > \(i - 1\)< / span > 阶或 < span class = "arithmatex" > \(i - 2\)< / span > 阶走来，因此 < span class = "arithmatex" > \(dp[i]\)< / span > 只可能等于 < span class = "arithmatex" > \(dp[i - 1] + cost[i]\)< / span > 或 < span class = "arithmatex" > \(dp[i - 2] + cost[i]\)< / span > 。为了尽可能减少代价，我们应该选择两者中较小的那一个，即：< / p >
< div class = "arithmatex" > \[
dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
\]< / div >
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< p > 这便可以引出「最优子结构」的含义：< strong > 原问题的最优解是从子问题的最优解构建得来的< / strong > 。< / p >
< p > 本题显然具有最优子结构：我们从两个子问题最优解 < span class = "arithmatex" > \(dp[i-1]\)< / span > , < span class = "arithmatex" > \(dp[i-2]\)< / span > 中挑选出较优的那一个，并用它构建出原问题 < span class = "arithmatex" > \(dp[i]\)< / span > 的最优解。< / p >
< p > 那么，上节的爬楼梯题目有没有最优子结构呢？它的目标是求解方案数量，看似是一个计数问题，但如果换一种问法：“求解最大方案数量”。我们意外地发现，< strong > 虽然题目修改前后是等价的，但最优子结构浮现出来了< / strong > ：第 < span class = "arithmatex" > \(n\)< / span > 阶最大方案数量等于第 < span class = "arithmatex" > \(n-1\)< / span > 阶和第 < span class = "arithmatex" > \(n-2\)< / span > 阶最大方案数量之和。所以说，最优子结构的解释方式比较灵活，在不同问题中会有不同的含义。< / p >
< p > 根据状态转移方程，以及初始状态 < span class = "arithmatex" > \(dp[1] = cost[1]\)< / span > , < span class = "arithmatex" > \(dp[2] = cost[2]\)< / span > ，可以得出动态规划代码。< / p >
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< div class = "tabbed-set tabbed-alternate" data-tabs = "1:12" > < input checked = "checked" id = "__tabbed_1_1" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_2" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_3" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_4" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_5" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_6" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_7" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_8" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_9" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_10" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_11" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_12" name = "__tabbed_1" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > Java< / label > < label for = "__tabbed_1_2" > C++< / label > < label for = "__tabbed_1_3" > Python< / label > < label for = "__tabbed_1_4" > Go< / label > < label for = "__tabbed_1_5" > JS< / label > < label for = "__tabbed_1_6" > TS< / label > < label for = "__tabbed_1_7" > C< / label > < label for = "__tabbed_1_8" > C#< / label > < label for = "__tabbed_1_9" > Swift< / label > < label for = "__tabbed_1_10" > Zig< / label > < label for = "__tabbed_1_11" > Dart< / label > < label for = "__tabbed_1_12" > Rust< / label > < / div >
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< div class = "tabbed-content" >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "na" > length< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
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< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-13" name = "__codelineno-0-13" href = "#__codelineno-0-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "na" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-14" name = "__codelineno-0-14" href = "#__codelineno-0-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-15" name = "__codelineno-0-15" href = "#__codelineno-0-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-16" name = "__codelineno-0-16" href = "#__codelineno-0-16" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > size< / span > < span class = "p" > ()< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-1-4" name = "__codelineno-1-4" href = "#__codelineno-1-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-1-5" name = "__codelineno-1-5" href = "#__codelineno-1-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-1-6" name = "__codelineno-1-6" href = "#__codelineno-1-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
2023-07-01 22:39:28 +08:00
< a id = "__codelineno-1-7" name = "__codelineno-1-7" href = "#__codelineno-1-7" > < / a > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-1-8" name = "__codelineno-1-8" href = "#__codelineno-1-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-1-9" name = "__codelineno-1-9" href = "#__codelineno-1-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-10" name = "__codelineno-1-10" href = "#__codelineno-1-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-11" name = "__codelineno-1-11" href = "#__codelineno-1-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-1-12" name = "__codelineno-1-12" href = "#__codelineno-1-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-13" name = "__codelineno-1-13" href = "#__codelineno-1-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-14" name = "__codelineno-1-14" href = "#__codelineno-1-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-1-15" name = "__codelineno-1-15" href = "#__codelineno-1-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-16" name = "__codelineno-1-16" href = "#__codelineno-1-16" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > min_cost_climbing_stairs_dp< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ])< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 爬楼梯最小代价：动态规划" " " < / span >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "n" > n< / span > < span class = "o" > =< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-2-4" name = "__codelineno-2-4" href = "#__codelineno-2-4" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 1< / span > < span class = "ow" > or< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-5" name = "__codelineno-2-5" href = "#__codelineno-2-5" > < / a > < span class = "k" > return< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-2-6" name = "__codelineno-2-6" href = "#__codelineno-2-6" > < / a > < span class = "c1" > # 初始化 dp 表，用于存储子问题的解< / span >
2023-07-01 22:39:28 +08:00
< a id = "__codelineno-2-7" name = "__codelineno-2-7" href = "#__codelineno-2-7" > < / a > < span class = "n" > dp< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-2-8" name = "__codelineno-2-8" href = "#__codelineno-2-8" > < / a > < span class = "c1" > # 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-2-9" name = "__codelineno-2-9" href = "#__codelineno-2-9" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-2-10" name = "__codelineno-2-10" href = "#__codelineno-2-10" > < / a > < span class = "c1" > # 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-2-11" name = "__codelineno-2-11" href = "#__codelineno-2-11" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-2-12" name = "__codelineno-2-12" href = "#__codelineno-2-12" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span > < span class = "o" > +< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-2-13" name = "__codelineno-2-13" href = "#__codelineno-2-13" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
2023-07-24 13:09:54 +08:00
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-3-2" name = "__codelineno-3-2" href = "#__codelineno-3-2" > < / a > < span class = "kd" > func< / span > < span class = "w" > < / span > < span class = "nx" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "nx" > cost< / span > < span class = "w" > < / span > < span class = "p" > []< / span > < span class = "kt" > int< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-3-3" name = "__codelineno-3-3" href = "#__codelineno-3-3" > < / a > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "nx" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-3-4" name = "__codelineno-3-4" href = "#__codelineno-3-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-3-5" name = "__codelineno-3-5" href = "#__codelineno-3-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "nx" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-3-6" name = "__codelineno-3-6" href = "#__codelineno-3-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-3-7" name = "__codelineno-3-7" href = "#__codelineno-3-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-3-8" name = "__codelineno-3-8" href = "#__codelineno-3-8" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nb" > make< / span > < span class = "p" > ([]< / span > < span class = "kt" > int< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-3-9" name = "__codelineno-3-9" href = "#__codelineno-3-9" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-3-10" name = "__codelineno-3-10" href = "#__codelineno-3-10" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-3-11" name = "__codelineno-3-11" href = "#__codelineno-3-11" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-3-12" name = "__codelineno-3-12" href = "#__codelineno-3-12" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-3-13" name = "__codelineno-3-13" href = "#__codelineno-3-13" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "o" > ++< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-3-14" name = "__codelineno-3-14" href = "#__codelineno-3-14" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nb" > int< / span > < span class = "p" > (< / span > < span class = "nx" > math< / span > < span class = "p" > .< / span > < span class = "nx" > Min< / span > < span class = "p" > (< / span > < span class = "nb" > float64< / span > < span class = "p" > (< / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ]),< / span > < span class = "w" > < / span > < span class = "nb" > float64< / span > < span class = "p" > (< / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "p" > ])))< / span >
< a id = "__codelineno-3-15" name = "__codelineno-3-15" href = "#__codelineno-3-15" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-3-16" name = "__codelineno-3-16" href = "#__codelineno-3-16" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-3-17" name = "__codelineno-3-17" href = "#__codelineno-3-17" > < / a > < span class = "p" > }< / span >
2023-07-01 22:39:28 +08:00
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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2023-07-03 16:55:46 +08:00
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > Length< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-7-4" name = "__codelineno-7-4" href = "#__codelineno-7-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-7-5" name = "__codelineno-7-5" href = "#__codelineno-7-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-7-6" name = "__codelineno-7-6" href = "#__codelineno-7-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
2023-07-03 16:55:46 +08:00
< a id = "__codelineno-7-7" name = "__codelineno-7-7" href = "#__codelineno-7-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-8" name = "__codelineno-7-8" href = "#__codelineno-7-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-7-9" name = "__codelineno-7-9" href = "#__codelineno-7-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-10" name = "__codelineno-7-10" href = "#__codelineno-7-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-11" name = "__codelineno-7-11" href = "#__codelineno-7-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-7-12" name = "__codelineno-7-12" href = "#__codelineno-7-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-13" name = "__codelineno-7-13" href = "#__codelineno-7-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-14" name = "__codelineno-7-14" href = "#__codelineno-7-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-7-15" name = "__codelineno-7-15" href = "#__codelineno-7-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-16" name = "__codelineno-7-16" href = "#__codelineno-7-16" > < / a > < span class = "p" > }< / span >
2023-07-01 22:39:28 +08:00
< / code > < / pre > < / div >
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2023-07-19 01:44:47 +08:00
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-8-1" name = "__codelineno-8-1" href = "#__codelineno-8-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-8-2" name = "__codelineno-8-2" href = "#__codelineno-8-2" > < / a > < span class = "kd" > func< / span > < span class = "nf" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > :< / span > < span class = "p" > [< / span > < span class = "nb" > Int< / span > < span class = "p" > ])< / span > < span class = "p" > -> < / span > < span class = "nb" > Int< / span > < span class = "p" > {< / span >
< a id = "__codelineno-8-3" name = "__codelineno-8-3" href = "#__codelineno-8-3" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n< / span > < span class = "p" > =< / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "bp" > count< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-8-4" name = "__codelineno-8-4" href = "#__codelineno-8-4" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "p" > ==< / span > < span class = "mi" > 1< / span > < span class = "o" > ||< / span > < span class = "n" > n< / span > < span class = "p" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > {< / span >
< a id = "__codelineno-8-5" name = "__codelineno-8-5" href = "#__codelineno-8-5" > < / a > < span class = "k" > return< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-8-6" name = "__codelineno-8-6" href = "#__codelineno-8-6" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-8-7" name = "__codelineno-8-7" href = "#__codelineno-8-7" > < / a > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-8-8" name = "__codelineno-8-8" href = "#__codelineno-8-8" > < / a > < span class = "kd" > var< / span > < span class = "nv" > dp< / span > < span class = "p" > =< / span > < span class = "nb" > Array< / span > < span class = "p" > (< / span > < span class = "n" > repeating< / span > < span class = "p" > :< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "bp" > count< / span > < span class = "p" > :< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-8-9" name = "__codelineno-8-9" href = "#__codelineno-8-9" > < / a > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-8-10" name = "__codelineno-8-10" href = "#__codelineno-8-10" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-8-11" name = "__codelineno-8-11" href = "#__codelineno-8-11" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "mi" > 2< / span >
< a id = "__codelineno-8-12" name = "__codelineno-8-12" href = "#__codelineno-8-12" > < / a > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-8-13" name = "__codelineno-8-13" href = "#__codelineno-8-13" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "k" > in< / span > < span class = "bp" > stride< / span > < span class = "p" > (< / span > < span class = "n" > from< / span > < span class = "p" > :< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > through< / span > < span class = "p" > :< / span > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "n" > by< / span > < span class = "p" > :< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "p" > {< / span >
< a id = "__codelineno-8-14" name = "__codelineno-8-14" href = "#__codelineno-8-14" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "bp" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span > < span class = "o" > +< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-8-15" name = "__codelineno-8-15" href = "#__codelineno-8-15" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-8-16" name = "__codelineno-8-16" href = "#__codelineno-8-16" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-8-17" name = "__codelineno-8-17" href = "#__codelineno-8-17" > < / a > < span class = "p" > }< / span >
2023-07-01 22:39:28 +08:00
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
2023-07-17 02:17:51 +08:00
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-9-1" name = "__codelineno-9-1" href = "#__codelineno-9-1" > < / a > < span class = "c1" > // 爬楼梯最小代价：动态规划< / span >
< a id = "__codelineno-9-2" name = "__codelineno-9-2" href = "#__codelineno-9-2" > < / a > < span class = "k" > fn< / span > < span class = "w" > < / span > < span class = "n" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "kr" > comptime< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "p" > []< / span > < span class = "kt" > i32< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "kt" > i32< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-9-3" name = "__codelineno-9-3" href = "#__codelineno-9-3" > < / a > < span class = "w" > < / span > < span class = "kr" > comptime< / span > < span class = "w" > < / span > < span class = "kr" > var< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > len< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-9-4" name = "__codelineno-9-4" href = "#__codelineno-9-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "k" > or< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-9-5" name = "__codelineno-9-5" href = "#__codelineno-9-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-9-6" name = "__codelineno-9-6" href = "#__codelineno-9-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-9-7" name = "__codelineno-9-7" href = "#__codelineno-9-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-9-8" name = "__codelineno-9-8" href = "#__codelineno-9-8" > < / a > < span class = "w" > < / span > < span class = "kr" > var< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "n" > _< / span > < span class = "p" > ]< / span > < span class = "kt" > i32< / span > < span class = "p" > {< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > }< / span > < span class = "w" > < / span > < span class = "o" > **< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-9-9" name = "__codelineno-9-9" href = "#__codelineno-9-9" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-9-10" name = "__codelineno-9-10" href = "#__codelineno-9-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-9-11" name = "__codelineno-9-11" href = "#__codelineno-9-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-9-12" name = "__codelineno-9-12" href = "#__codelineno-9-12" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-9-13" name = "__codelineno-9-13" href = "#__codelineno-9-13" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ..< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "n" > i< / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-9-14" name = "__codelineno-9-14" href = "#__codelineno-9-14" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > @min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-9-15" name = "__codelineno-9-15" href = "#__codelineno-9-15" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-9-16" name = "__codelineno-9-16" href = "#__codelineno-9-16" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-9-17" name = "__codelineno-9-17" href = "#__codelineno-9-17" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-10-1" name = "__codelineno-10-1" href = "#__codelineno-10-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
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< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.rs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-11-1" name = "__codelineno-11-1" href = "#__codelineno-11-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-11-2" name = "__codelineno-11-2" href = "#__codelineno-11-2" > < / a > < span class = "k" > fn< / span > < span class = "nf" > min_cost_climbing_stairs_dp< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > : < span class = "kp" > & < / span > < span class = "p" > [< / span > < span class = "kt" > i32< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > -> < span class = "kt" > i32< / span > < span class = "p" > {< / span >
< a id = "__codelineno-11-3" name = "__codelineno-11-3" href = "#__codelineno-11-3" > < / a > < span class = "w" > < / span > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > len< / span > < span class = "p" > ()< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-4" name = "__codelineno-11-4" href = "#__codelineno-11-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-11-5" name = "__codelineno-11-5" href = "#__codelineno-11-5" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-11-6" name = "__codelineno-11-6" href = "#__codelineno-11-6" > < / a > < span class = "w" > < / span > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "k" > mut< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "fm" > vec!< / span > < span class = "p" > [< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-11-7" name = "__codelineno-11-7" href = "#__codelineno-11-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-11-8" name = "__codelineno-11-8" href = "#__codelineno-11-8" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-11-9" name = "__codelineno-11-9" href = "#__codelineno-11-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-11-10" name = "__codelineno-11-10" href = "#__codelineno-11-10" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-11-11" name = "__codelineno-11-11" href = "#__codelineno-11-11" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "k" > in< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "o" > ..=< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-11-12" name = "__codelineno-11-12" href = "#__codelineno-11-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cmp< / span > ::< span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-11-13" name = "__codelineno-11-13" href = "#__codelineno-11-13" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-11-14" name = "__codelineno-11-14" href = "#__codelineno-11-14" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-11-15" name = "__codelineno-11-15" href = "#__codelineno-11-15" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
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< / div >
< / div >
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< p > < img alt = "爬楼梯最小代价的动态规划过程" src = "../dp_problem_features.assets/min_cost_cs_dp.png" / > < / p >
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< p align = "center" > Fig. 爬楼梯最小代价的动态规划过程 < / p >
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< p > 本题也可以进行状态压缩，将一维压缩至零维，使得空间复杂度从 < span class = "arithmatex" > \(O(n)\)< / span > 降低至 < span class = "arithmatex" > \(O(1)\)< / span > 。< / p >
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< div class = "tabbed-set tabbed-alternate" data-tabs = "2:12" > < input checked = "checked" id = "__tabbed_2_1" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_2" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_3" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_4" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_5" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_6" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_7" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_8" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_9" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_10" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_11" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_12" name = "__tabbed_2" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_2_1" > Java< / label > < label for = "__tabbed_2_2" > C++< / label > < label for = "__tabbed_2_3" > Python< / label > < label for = "__tabbed_2_4" > Go< / label > < label for = "__tabbed_2_5" > JS< / label > < label for = "__tabbed_2_6" > TS< / label > < label for = "__tabbed_2_7" > C< / label > < label for = "__tabbed_2_8" > C#< / label > < label for = "__tabbed_2_9" > Swift< / label > < label for = "__tabbed_2_10" > Zig< / label > < label for = "__tabbed_2_11" > Dart< / label > < label for = "__tabbed_2_12" > Rust< / label > < / div >
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< div class = "tabbed-content" >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-12-1" name = "__codelineno-12-1" href = "#__codelineno-12-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-12-2" name = "__codelineno-12-2" href = "#__codelineno-12-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDPComp< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-12-3" name = "__codelineno-12-3" href = "#__codelineno-12-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "na" > length< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-12-4" name = "__codelineno-12-4" href = "#__codelineno-12-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
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< a id = "__codelineno-12-5" name = "__codelineno-12-5" href = "#__codelineno-12-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-12-6" name = "__codelineno-12-6" href = "#__codelineno-12-6" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-12-7" name = "__codelineno-12-7" href = "#__codelineno-12-7" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-12-8" name = "__codelineno-12-8" href = "#__codelineno-12-8" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-12-9" name = "__codelineno-12-9" href = "#__codelineno-12-9" > < / a > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "na" > min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-12-10" name = "__codelineno-12-10" href = "#__codelineno-12-10" > < / a > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-12-11" name = "__codelineno-12-11" href = "#__codelineno-12-11" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-12-12" name = "__codelineno-12-12" href = "#__codelineno-12-12" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-12-13" name = "__codelineno-12-13" href = "#__codelineno-12-13" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-13-1" name = "__codelineno-13-1" href = "#__codelineno-13-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-13-2" name = "__codelineno-13-2" href = "#__codelineno-13-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDPComp< / span > < span class = "p" > (< / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-13-3" name = "__codelineno-13-3" href = "#__codelineno-13-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > size< / span > < span class = "p" > ()< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-13-4" name = "__codelineno-13-4" href = "#__codelineno-13-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-5" name = "__codelineno-13-5" href = "#__codelineno-13-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-13-6" name = "__codelineno-13-6" href = "#__codelineno-13-6" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-13-7" name = "__codelineno-13-7" href = "#__codelineno-13-7" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-13-8" name = "__codelineno-13-8" href = "#__codelineno-13-8" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-13-9" name = "__codelineno-13-9" href = "#__codelineno-13-9" > < / a > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-13-10" name = "__codelineno-13-10" href = "#__codelineno-13-10" > < / a > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-13-11" name = "__codelineno-13-11" href = "#__codelineno-13-11" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-13-12" name = "__codelineno-13-12" href = "#__codelineno-13-12" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-13-13" name = "__codelineno-13-13" href = "#__codelineno-13-13" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-14-1" name = "__codelineno-14-1" href = "#__codelineno-14-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > min_cost_climbing_stairs_dp_comp< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ])< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-14-2" name = "__codelineno-14-2" href = "#__codelineno-14-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 爬楼梯最小代价：状态压缩后的动态规划" " " < / span >
< a id = "__codelineno-14-3" name = "__codelineno-14-3" href = "#__codelineno-14-3" > < / a > < span class = "n" > n< / span > < span class = "o" > =< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-14-4" name = "__codelineno-14-4" href = "#__codelineno-14-4" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 1< / span > < span class = "ow" > or< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > :< / span >
< a id = "__codelineno-14-5" name = "__codelineno-14-5" href = "#__codelineno-14-5" > < / a > < span class = "k" > return< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-14-6" name = "__codelineno-14-6" href = "#__codelineno-14-6" > < / a > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "o" > =< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-14-7" name = "__codelineno-14-7" href = "#__codelineno-14-7" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-14-8" name = "__codelineno-14-8" href = "#__codelineno-14-8" > < / a > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "o" > =< / span > < span class = "n" > b< / span > < span class = "p" > ,< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-14-9" name = "__codelineno-14-9" href = "#__codelineno-14-9" > < / a > < span class = "k" > return< / span > < span class = "n" > b< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-15-1" name = "__codelineno-15-1" href = "#__codelineno-15-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-15-2" name = "__codelineno-15-2" href = "#__codelineno-15-2" > < / a > < span class = "kd" > func< / span > < span class = "w" > < / span > < span class = "nx" > minCostClimbingStairsDPComp< / span > < span class = "p" > (< / span > < span class = "nx" > cost< / span > < span class = "w" > < / span > < span class = "p" > []< / span > < span class = "kt" > int< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-15-3" name = "__codelineno-15-3" href = "#__codelineno-15-3" > < / a > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "nx" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-15-4" name = "__codelineno-15-4" href = "#__codelineno-15-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-15-5" name = "__codelineno-15-5" href = "#__codelineno-15-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "nx" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-15-6" name = "__codelineno-15-6" href = "#__codelineno-15-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-15-7" name = "__codelineno-15-7" href = "#__codelineno-15-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-15-8" name = "__codelineno-15-8" href = "#__codelineno-15-8" > < / a > < span class = "w" > < / span > < span class = "nx" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > b< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-15-9" name = "__codelineno-15-9" href = "#__codelineno-15-9" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-15-10" name = "__codelineno-15-10" href = "#__codelineno-15-10" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "o" > ++< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-15-11" name = "__codelineno-15-11" href = "#__codelineno-15-11" > < / a > < span class = "w" > < / span > < span class = "nx" > tmp< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > b< / span >
< a id = "__codelineno-15-12" name = "__codelineno-15-12" href = "#__codelineno-15-12" > < / a > < span class = "w" > < / span > < span class = "nx" > b< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nb" > int< / span > < span class = "p" > (< / span > < span class = "nx" > math< / span > < span class = "p" > .< / span > < span class = "nx" > Min< / span > < span class = "p" > (< / span > < span class = "nb" > float64< / span > < span class = "p" > (< / span > < span class = "nx" > a< / span > < span class = "p" > ),< / span > < span class = "w" > < / span > < span class = "nb" > float64< / span > < span class = "p" > (< / span > < span class = "nx" > tmp< / span > < span class = "o" > +< / span > < span class = "nx" > cost< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "p" > ])))< / span >
< a id = "__codelineno-15-13" name = "__codelineno-15-13" href = "#__codelineno-15-13" > < / a > < span class = "w" > < / span > < span class = "nx" > a< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > tmp< / span >
< a id = "__codelineno-15-14" name = "__codelineno-15-14" href = "#__codelineno-15-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-15-15" name = "__codelineno-15-15" href = "#__codelineno-15-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "nx" > b< / span >
< a id = "__codelineno-15-16" name = "__codelineno-15-16" href = "#__codelineno-15-16" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-16-1" name = "__codelineno-16-1" href = "#__codelineno-16-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-17-1" name = "__codelineno-17-1" href = "#__codelineno-17-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-18-1" name = "__codelineno-18-1" href = "#__codelineno-18-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-19-1" name = "__codelineno-19-1" href = "#__codelineno-19-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-19-2" name = "__codelineno-19-2" href = "#__codelineno-19-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDPComp< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-19-3" name = "__codelineno-19-3" href = "#__codelineno-19-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > Length< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-19-4" name = "__codelineno-19-4" href = "#__codelineno-19-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-19-5" name = "__codelineno-19-5" href = "#__codelineno-19-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-19-6" name = "__codelineno-19-6" href = "#__codelineno-19-6" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-19-7" name = "__codelineno-19-7" href = "#__codelineno-19-7" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-19-8" name = "__codelineno-19-8" href = "#__codelineno-19-8" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-19-9" name = "__codelineno-19-9" href = "#__codelineno-19-9" > < / a > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-19-10" name = "__codelineno-19-10" href = "#__codelineno-19-10" > < / a > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-19-11" name = "__codelineno-19-11" href = "#__codelineno-19-11" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-19-12" name = "__codelineno-19-12" href = "#__codelineno-19-12" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-19-13" name = "__codelineno-19-13" href = "#__codelineno-19-13" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-20-1" name = "__codelineno-20-1" href = "#__codelineno-20-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-20-2" name = "__codelineno-20-2" href = "#__codelineno-20-2" > < / a > < span class = "kd" > func< / span > < span class = "nf" > minCostClimbingStairsDPComp< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > :< / span > < span class = "p" > [< / span > < span class = "nb" > Int< / span > < span class = "p" > ])< / span > < span class = "p" > -> < / span > < span class = "nb" > Int< / span > < span class = "p" > {< / span >
< a id = "__codelineno-20-3" name = "__codelineno-20-3" href = "#__codelineno-20-3" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n< / span > < span class = "p" > =< / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "bp" > count< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-20-4" name = "__codelineno-20-4" href = "#__codelineno-20-4" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "p" > ==< / span > < span class = "mi" > 1< / span > < span class = "o" > ||< / span > < span class = "n" > n< / span > < span class = "p" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > {< / span >
< a id = "__codelineno-20-5" name = "__codelineno-20-5" href = "#__codelineno-20-5" > < / a > < span class = "k" > return< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-20-6" name = "__codelineno-20-6" href = "#__codelineno-20-6" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-20-7" name = "__codelineno-20-7" href = "#__codelineno-20-7" > < / a > < span class = "kd" > var< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > )< / span > < span class = "p" > =< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span >
< a id = "__codelineno-20-8" name = "__codelineno-20-8" href = "#__codelineno-20-8" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "k" > in< / span > < span class = "bp" > stride< / span > < span class = "p" > (< / span > < span class = "n" > from< / span > < span class = "p" > :< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > through< / span > < span class = "p" > :< / span > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "n" > by< / span > < span class = "p" > :< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "p" > {< / span >
< a id = "__codelineno-20-9" name = "__codelineno-20-9" href = "#__codelineno-20-9" > < / a > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > )< / span > < span class = "p" > =< / span > < span class = "p" > (< / span > < span class = "n" > b< / span > < span class = "p" > ,< / span > < span class = "bp" > min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ])< / span >
< a id = "__codelineno-20-10" name = "__codelineno-20-10" href = "#__codelineno-20-10" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-20-11" name = "__codelineno-20-11" href = "#__codelineno-20-11" > < / a > < span class = "k" > return< / span > < span class = "n" > b< / span >
< a id = "__codelineno-20-12" name = "__codelineno-20-12" href = "#__codelineno-20-12" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-21-1" name = "__codelineno-21-1" href = "#__codelineno-21-1" > < / a > < span class = "c1" > // 爬楼梯最小代价：状态压缩后的动态规划< / span >
< a id = "__codelineno-21-2" name = "__codelineno-21-2" href = "#__codelineno-21-2" > < / a > < span class = "k" > fn< / span > < span class = "w" > < / span > < span class = "n" > minCostClimbingStairsDPComp< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "p" > []< / span > < span class = "kt" > i32< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "kt" > i32< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-21-3" name = "__codelineno-21-3" href = "#__codelineno-21-3" > < / a > < span class = "w" > < / span > < span class = "kr" > var< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > len< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-4" name = "__codelineno-21-4" href = "#__codelineno-21-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "k" > or< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-21-5" name = "__codelineno-21-5" href = "#__codelineno-21-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-21-6" name = "__codelineno-21-6" href = "#__codelineno-21-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-21-7" name = "__codelineno-21-7" href = "#__codelineno-21-7" > < / a > < span class = "w" > < / span > < span class = "kr" > var< / span > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-21-8" name = "__codelineno-21-8" href = "#__codelineno-21-8" > < / a > < span class = "w" > < / span > < span class = "kr" > var< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-21-9" name = "__codelineno-21-9" href = "#__codelineno-21-9" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-21-10" name = "__codelineno-21-10" href = "#__codelineno-21-10" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ..< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "n" > i< / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-21-11" name = "__codelineno-21-11" href = "#__codelineno-21-11" > < / a > < span class = "w" > < / span > < span class = "kr" > var< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-12" name = "__codelineno-21-12" href = "#__codelineno-21-12" > < / a > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > @min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-21-13" name = "__codelineno-21-13" href = "#__codelineno-21-13" > < / a > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-14" name = "__codelineno-21-14" href = "#__codelineno-21-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-21-15" name = "__codelineno-21-15" href = "#__codelineno-21-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-16" name = "__codelineno-21-16" href = "#__codelineno-21-16" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-22-1" name = "__codelineno-22-1" href = "#__codelineno-22-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.rs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-23-1" name = "__codelineno-23-1" href = "#__codelineno-23-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-23-2" name = "__codelineno-23-2" href = "#__codelineno-23-2" > < / a > < span class = "k" > fn< / span > < span class = "nf" > min_cost_climbing_stairs_dp_comp< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > : < span class = "kp" > & < / span > < span class = "p" > [< / span > < span class = "kt" > i32< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > -> < span class = "kt" > i32< / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-3" name = "__codelineno-23-3" href = "#__codelineno-23-3" > < / a > < span class = "w" > < / span > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > len< / span > < span class = "p" > ()< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-4" name = "__codelineno-23-4" href = "#__codelineno-23-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > };< / span >
< a id = "__codelineno-23-5" name = "__codelineno-23-5" href = "#__codelineno-23-5" > < / a > < span class = "w" > < / span > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "k" > mut< / span > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > mut< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]);< / span >
< a id = "__codelineno-23-6" name = "__codelineno-23-6" href = "#__codelineno-23-6" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "k" > in< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "o" > ..=< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-7" name = "__codelineno-23-7" href = "#__codelineno-23-7" > < / a > < span class = "w" > < / span > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-8" name = "__codelineno-23-8" href = "#__codelineno-23-8" > < / a > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cmp< / span > ::< span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-9" name = "__codelineno-23-9" href = "#__codelineno-23-9" > < / a > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-10" name = "__codelineno-23-10" href = "#__codelineno-23-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-23-11" name = "__codelineno-23-11" href = "#__codelineno-23-11" > < / a > < span class = "w" > < / span > < span class = "n" > b< / span >
< a id = "__codelineno-23-12" name = "__codelineno-23-12" href = "#__codelineno-23-12" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
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< h2 id = "1422" > 14.2.2. 无后效性< a class = "headerlink" href = "#1422" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 「无后效性」是动态规划能够有效解决问题的重要特性之一，定义为：< strong > 给定一个确定的状态，它的未来发展只与当前状态有关，而与当前状态过去所经历过的所有状态无关< / strong > 。< / p >
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< p > 以爬楼梯问题为例，给定状态 < span class = "arithmatex" > \(i\)< / span > ，它会发展出状态 < span class = "arithmatex" > \(i+1\)< / span > 和状态 < span class = "arithmatex" > \(i+2\)< / span > ，分别对应跳 < span class = "arithmatex" > \(1\)< / span > 步和跳 < span class = "arithmatex" > \(2\)< / span > 步。在做出这两种选择时，我们无需考虑状态 < span class = "arithmatex" > \(i\)< / span > 之前的状态，它们对状态 < span class = "arithmatex" > \(i\)< / span > 的未来没有影响。< / p >
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< p > 然而，如果我们向爬楼梯问题添加一个约束，情况就不一样了。< / p >
< div class = "admonition question" >
< p class = "admonition-title" > 带约束爬楼梯< / p >
< p > 给定一个共有 < span class = "arithmatex" > \(n\)< / span > 阶的楼梯，你每步可以上 < span class = "arithmatex" > \(1\)< / span > 阶或者 < span class = "arithmatex" > \(2\)< / span > 阶，< strong > 但不能连续两轮跳 < span class = "arithmatex" > \(1\)< / span > 阶< / strong > ，请问有多少种方案可以爬到楼顶。< / p >
< / div >
< p > 例如，爬上第 < span class = "arithmatex" > \(3\)< / span > 阶仅剩 < span class = "arithmatex" > \(2\)< / span > 种可行方案，其中连续三次跳 < span class = "arithmatex" > \(1\)< / span > 阶的方案不满足约束条件，因此被舍弃。< / p >
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< p > < img alt = "带约束爬到第 3 阶的方案数量" src = "../dp_problem_features.assets/climbing_stairs_constraint_example.png" / > < / p >
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< p align = "center" > Fig. 带约束爬到第 3 阶的方案数量 < / p >
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< p > 在该问题中，如果上一轮是跳 < span class = "arithmatex" > \(1\)< / span > 阶上来的，那么下一轮就必须跳 < span class = "arithmatex" > \(2\)< / span > 阶。这意味着，< strong > 下一步选择不能由当前状态（当前楼梯阶数）独立决定，还和前一个状态（上轮楼梯阶数）有关< / strong > 。< / p >
< p > 不难发现，此问题已不满足无后效性，状态转移方程 < span class = "arithmatex" > \(dp[i] = dp[i-1] + dp[i-2]\)< / span > 也失效了，因为 < span class = "arithmatex" > \(dp[i-1]\)< / span > 代表本轮跳 < span class = "arithmatex" > \(1\)< / span > 阶，但其中包含了许多“上一轮跳 < span class = "arithmatex" > \(1\)< / span > 阶上来的”方案，而为了满足约束，我们就不能将 < span class = "arithmatex" > \(dp[i-1]\)< / span > 直接计入 < span class = "arithmatex" > \(dp[i]\)< / span > 中。< / p >
< p > 为此，我们需要扩展状态定义：< strong > 状态 < span class = "arithmatex" > \([i, j]\)< / span > 表示处在第 < span class = "arithmatex" > \(i\)< / span > 阶、并且上一轮跳了 < span class = "arithmatex" > \(j\)< / span > 阶< / strong > ，其中 < span class = "arithmatex" > \(j \in \{1, 2\}\)< / span > 。此状态定义有效地区分了上一轮跳了 < span class = "arithmatex" > \(1\)< / span > 阶还是 < span class = "arithmatex" > \(2\)< / span > 阶，我们可以据此来决定下一步该怎么跳：< / p >
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< ul >
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< li > 当 < span class = "arithmatex" > \(j\)< / span > 等于 < span class = "arithmatex" > \(1\)< / span > ，即上一轮跳了 < span class = "arithmatex" > \(1\)< / span > 阶时，这一轮只能选择跳 < span class = "arithmatex" > \(2\)< / span > 阶。< / li >
< li > 当 < span class = "arithmatex" > \(j\)< / span > 等于 < span class = "arithmatex" > \(2\)< / span > ，即上一轮跳了 < span class = "arithmatex" > \(2\)< / span > 阶时，这一轮可选择跳 < span class = "arithmatex" > \(1\)< / span > 阶或跳 < span class = "arithmatex" > \(2\)< / span > 阶。< / li >
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< / ul >
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< p > 在该定义下，< span class = "arithmatex" > \(dp[i, j]\)< / span > 表示状态 < span class = "arithmatex" > \([i, j]\)< / span > 对应的方案数。在该定义下的状态转移方程为：< / p >
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< div class = "arithmatex" > \[
\begin{cases}
dp[i, 1] = dp[i-1, 2] \\
dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
\end{cases}
\]< / div >
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< p > < img alt = "考虑约束下的递推关系" src = "../dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png" / > < / p >
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< p align = "center" > Fig. 考虑约束下的递推关系 < / p >
< p > 最终，返回 < span class = "arithmatex" > \(dp[n, 1] + dp[n, 2]\)< / span > 即可，两者之和代表爬到第 < span class = "arithmatex" > \(n\)< / span > 阶的方案总数。< / p >
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< div class = "tabbed-set tabbed-alternate" data-tabs = "3:12" > < input checked = "checked" id = "__tabbed_3_1" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_2" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_3" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_4" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_5" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_6" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_7" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_8" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_9" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_10" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_11" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_12" name = "__tabbed_3" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_3_1" > Java< / label > < label for = "__tabbed_3_2" > C++< / label > < label for = "__tabbed_3_3" > Python< / label > < label for = "__tabbed_3_4" > Go< / label > < label for = "__tabbed_3_5" > JS< / label > < label for = "__tabbed_3_6" > TS< / label > < label for = "__tabbed_3_7" > C< / label > < label for = "__tabbed_3_8" > C#< / label > < label for = "__tabbed_3_9" > Swift< / label > < label for = "__tabbed_3_10" > Zig< / label > < label for = "__tabbed_3_11" > Dart< / label > < label for = "__tabbed_3_12" > Rust< / label > < / div >
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< div class = "tabbed-content" >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-24-1" name = "__codelineno-24-1" href = "#__codelineno-24-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-24-2" name = "__codelineno-24-2" href = "#__codelineno-24-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-24-3" name = "__codelineno-24-3" href = "#__codelineno-24-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-24-4" name = "__codelineno-24-4" href = "#__codelineno-24-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-5" name = "__codelineno-24-5" href = "#__codelineno-24-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-24-6" name = "__codelineno-24-6" href = "#__codelineno-24-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-24-7" name = "__codelineno-24-7" href = "#__codelineno-24-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [][]< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 3< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-8" name = "__codelineno-24-8" href = "#__codelineno-24-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-24-9" name = "__codelineno-24-9" href = "#__codelineno-24-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-10" name = "__codelineno-24-10" href = "#__codelineno-24-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-11" name = "__codelineno-24-11" href = "#__codelineno-24-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-12" name = "__codelineno-24-12" href = "#__codelineno-24-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-13" name = "__codelineno-24-13" href = "#__codelineno-24-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-24-14" name = "__codelineno-24-14" href = "#__codelineno-24-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-24-15" name = "__codelineno-24-15" href = "#__codelineno-24-15" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-16" name = "__codelineno-24-16" href = "#__codelineno-24-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-17" name = "__codelineno-24-17" href = "#__codelineno-24-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-24-18" name = "__codelineno-24-18" href = "#__codelineno-24-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-19" name = "__codelineno-24-19" href = "#__codelineno-24-19" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-25-1" name = "__codelineno-25-1" href = "#__codelineno-25-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-25-2" name = "__codelineno-25-2" href = "#__codelineno-25-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-25-3" name = "__codelineno-25-3" href = "#__codelineno-25-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-25-4" name = "__codelineno-25-4" href = "#__codelineno-25-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-25-5" name = "__codelineno-25-5" href = "#__codelineno-25-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-25-6" name = "__codelineno-25-6" href = "#__codelineno-25-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
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< a id = "__codelineno-25-7" name = "__codelineno-25-7" href = "#__codelineno-25-7" > < / a > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ));< / span >
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< a id = "__codelineno-25-8" name = "__codelineno-25-8" href = "#__codelineno-25-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
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< a id = "__codelineno-25-9" name = "__codelineno-25-9" href = "#__codelineno-25-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-25-10" name = "__codelineno-25-10" href = "#__codelineno-25-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-25-11" name = "__codelineno-25-11" href = "#__codelineno-25-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-25-12" name = "__codelineno-25-12" href = "#__codelineno-25-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-25-13" name = "__codelineno-25-13" href = "#__codelineno-25-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
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< a id = "__codelineno-25-14" name = "__codelineno-25-14" href = "#__codelineno-25-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-25-15" name = "__codelineno-25-15" href = "#__codelineno-25-15" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-25-16" name = "__codelineno-25-16" href = "#__codelineno-25-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
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< a id = "__codelineno-25-17" name = "__codelineno-25-17" href = "#__codelineno-25-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
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< a id = "__codelineno-25-18" name = "__codelineno-25-18" href = "#__codelineno-25-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
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< a id = "__codelineno-25-19" name = "__codelineno-25-19" href = "#__codelineno-25-19" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-26-1" name = "__codelineno-26-1" href = "#__codelineno-26-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > climbing_stairs_constraint_dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-26-2" name = "__codelineno-26-2" href = "#__codelineno-26-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 带约束爬楼梯：动态规划" " " < / span >
< a id = "__codelineno-26-3" name = "__codelineno-26-3" href = "#__codelineno-26-3" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 1< / span > < span class = "ow" > or< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > :< / span >
< a id = "__codelineno-26-4" name = "__codelineno-26-4" href = "#__codelineno-26-4" > < / a > < span class = "k" > return< / span > < span class = "n" > n< / span >
< a id = "__codelineno-26-5" name = "__codelineno-26-5" href = "#__codelineno-26-5" > < / a > < span class = "c1" > # 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-26-6" name = "__codelineno-26-6" href = "#__codelineno-26-6" > < / a > < span class = "n" > dp< / span > < span class = "o" > =< / span > < span class = "p" > [[< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "mi" > 3< / span > < span class = "k" > for< / span > < span class = "n" > _< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )]< / span >
< a id = "__codelineno-26-7" name = "__codelineno-26-7" href = "#__codelineno-26-7" > < / a > < span class = "c1" > # 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-26-8" name = "__codelineno-26-8" href = "#__codelineno-26-8" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-26-9" name = "__codelineno-26-9" href = "#__codelineno-26-9" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-26-10" name = "__codelineno-26-10" href = "#__codelineno-26-10" > < / a > < span class = "c1" > # 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-26-11" name = "__codelineno-26-11" href = "#__codelineno-26-11" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-26-12" name = "__codelineno-26-12" href = "#__codelineno-26-12" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-26-13" name = "__codelineno-26-13" href = "#__codelineno-26-13" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-26-14" name = "__codelineno-26-14" href = "#__codelineno-26-14" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-27-1" name = "__codelineno-27-1" href = "#__codelineno-27-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-27-2" name = "__codelineno-27-2" href = "#__codelineno-27-2" > < / a > < span class = "kd" > func< / span > < span class = "w" > < / span > < span class = "nx" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-27-3" name = "__codelineno-27-3" href = "#__codelineno-27-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-27-4" name = "__codelineno-27-4" href = "#__codelineno-27-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "nx" > n< / span >
< a id = "__codelineno-27-5" name = "__codelineno-27-5" href = "#__codelineno-27-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-27-6" name = "__codelineno-27-6" href = "#__codelineno-27-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-27-7" name = "__codelineno-27-7" href = "#__codelineno-27-7" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nb" > make< / span > < span class = "p" > ([][< / span > < span class = "mi" > 3< / span > < span class = "p" > ]< / span > < span class = "kt" > int< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-27-8" name = "__codelineno-27-8" href = "#__codelineno-27-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-27-9" name = "__codelineno-27-9" href = "#__codelineno-27-9" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-27-10" name = "__codelineno-27-10" href = "#__codelineno-27-10" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-27-11" name = "__codelineno-27-11" href = "#__codelineno-27-11" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-27-12" name = "__codelineno-27-12" href = "#__codelineno-27-12" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-27-13" name = "__codelineno-27-13" href = "#__codelineno-27-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-27-14" name = "__codelineno-27-14" href = "#__codelineno-27-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "nx" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "nx" > i< / span > < span class = "o" > ++< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-27-15" name = "__codelineno-27-15" href = "#__codelineno-27-15" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-27-16" name = "__codelineno-27-16" href = "#__codelineno-27-16" > < / a > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-27-17" name = "__codelineno-27-17" href = "#__codelineno-27-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-27-18" name = "__codelineno-27-18" href = "#__codelineno-27-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "nx" > dp< / span > < span class = "p" > [< / span > < span class = "nx" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-27-19" name = "__codelineno-27-19" href = "#__codelineno-27-19" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-28-1" name = "__codelineno-28-1" href = "#__codelineno-28-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-29-1" name = "__codelineno-29-1" href = "#__codelineno-29-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-30-1" name = "__codelineno-30-1" href = "#__codelineno-30-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-31-1" name = "__codelineno-31-1" href = "#__codelineno-31-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-31-2" name = "__codelineno-31-2" href = "#__codelineno-31-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-31-3" name = "__codelineno-31-3" href = "#__codelineno-31-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-31-4" name = "__codelineno-31-4" href = "#__codelineno-31-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-31-5" name = "__codelineno-31-5" href = "#__codelineno-31-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-31-6" name = "__codelineno-31-6" href = "#__codelineno-31-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
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< a id = "__codelineno-31-7" name = "__codelineno-31-7" href = "#__codelineno-31-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [,]< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ];< / span >
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< a id = "__codelineno-31-8" name = "__codelineno-31-8" href = "#__codelineno-31-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
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< a id = "__codelineno-31-9" name = "__codelineno-31-9" href = "#__codelineno-31-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-31-10" name = "__codelineno-31-10" href = "#__codelineno-31-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-31-11" name = "__codelineno-31-11" href = "#__codelineno-31-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-31-12" name = "__codelineno-31-12" href = "#__codelineno-31-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-31-13" name = "__codelineno-31-13" href = "#__codelineno-31-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
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< a id = "__codelineno-31-14" name = "__codelineno-31-14" href = "#__codelineno-31-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-31-15" name = "__codelineno-31-15" href = "#__codelineno-31-15" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-31-16" name = "__codelineno-31-16" href = "#__codelineno-31-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
2023-07-17 02:17:51 +08:00
< a id = "__codelineno-31-17" name = "__codelineno-31-17" href = "#__codelineno-31-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
2023-07-26 10:57:52 +08:00
< a id = "__codelineno-31-18" name = "__codelineno-31-18" href = "#__codelineno-31-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
2023-07-17 02:17:51 +08:00
< a id = "__codelineno-31-19" name = "__codelineno-31-19" href = "#__codelineno-31-19" > < / a > < span class = "p" > }< / span >
2023-07-01 22:39:28 +08:00
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
2023-07-26 10:57:52 +08:00
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-32-1" name = "__codelineno-32-1" href = "#__codelineno-32-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-32-2" name = "__codelineno-32-2" href = "#__codelineno-32-2" > < / a > < span class = "kd" > func< / span > < span class = "nf" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "p" > :< / span > < span class = "nb" > Int< / span > < span class = "p" > )< / span > < span class = "p" > -> < / span > < span class = "nb" > Int< / span > < span class = "p" > {< / span >
< a id = "__codelineno-32-3" name = "__codelineno-32-3" href = "#__codelineno-32-3" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "p" > ==< / span > < span class = "mi" > 1< / span > < span class = "o" > ||< / span > < span class = "n" > n< / span > < span class = "p" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > {< / span >
< a id = "__codelineno-32-4" name = "__codelineno-32-4" href = "#__codelineno-32-4" > < / a > < span class = "k" > return< / span > < span class = "n" > n< / span >
< a id = "__codelineno-32-5" name = "__codelineno-32-5" href = "#__codelineno-32-5" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-32-6" name = "__codelineno-32-6" href = "#__codelineno-32-6" > < / a > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-32-7" name = "__codelineno-32-7" href = "#__codelineno-32-7" > < / a > < span class = "kd" > var< / span > < span class = "nv" > dp< / span > < span class = "p" > =< / span > < span class = "nb" > Array< / span > < span class = "p" > (< / span > < span class = "n" > repeating< / span > < span class = "p" > :< / span > < span class = "nb" > Array< / span > < span class = "p" > (< / span > < span class = "n" > repeating< / span > < span class = "p" > :< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "bp" > count< / span > < span class = "p" > :< / span > < span class = "mi" > 3< / span > < span class = "p" > ),< / span > < span class = "bp" > count< / span > < span class = "p" > :< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-32-8" name = "__codelineno-32-8" href = "#__codelineno-32-8" > < / a > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-32-9" name = "__codelineno-32-9" href = "#__codelineno-32-9" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-32-10" name = "__codelineno-32-10" href = "#__codelineno-32-10" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-32-11" name = "__codelineno-32-11" href = "#__codelineno-32-11" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-32-12" name = "__codelineno-32-12" href = "#__codelineno-32-12" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-32-13" name = "__codelineno-32-13" href = "#__codelineno-32-13" > < / a > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-32-14" name = "__codelineno-32-14" href = "#__codelineno-32-14" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "k" > in< / span > < span class = "bp" > stride< / span > < span class = "p" > (< / span > < span class = "n" > from< / span > < span class = "p" > :< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > through< / span > < span class = "p" > :< / span > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "n" > by< / span > < span class = "p" > :< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "p" > {< / span >
< a id = "__codelineno-32-15" name = "__codelineno-32-15" href = "#__codelineno-32-15" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-32-16" name = "__codelineno-32-16" href = "#__codelineno-32-16" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "p" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-32-17" name = "__codelineno-32-17" href = "#__codelineno-32-17" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-32-18" name = "__codelineno-32-18" href = "#__codelineno-32-18" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-32-19" name = "__codelineno-32-19" href = "#__codelineno-32-19" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-33-1" name = "__codelineno-33-1" href = "#__codelineno-33-1" > < / a > < span class = "c1" > // 带约束爬楼梯：动态规划< / span >
< a id = "__codelineno-33-2" name = "__codelineno-33-2" href = "#__codelineno-33-2" > < / a > < span class = "k" > fn< / span > < span class = "w" > < / span > < span class = "n" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "kr" > comptime< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > usize< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "kt" > i32< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-33-3" name = "__codelineno-33-3" href = "#__codelineno-33-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "k" > or< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-33-4" name = "__codelineno-33-4" href = "#__codelineno-33-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "nb" > @intCast< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "p" > );< / span >
< a id = "__codelineno-33-5" name = "__codelineno-33-5" href = "#__codelineno-33-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-33-6" name = "__codelineno-33-6" href = "#__codelineno-33-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-33-7" name = "__codelineno-33-7" href = "#__codelineno-33-7" > < / a > < span class = "w" > < / span > < span class = "kr" > var< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "n" > _< / span > < span class = "p" > ][< / span > < span class = "mi" > 3< / span > < span class = "p" > ]< / span > < span class = "kt" > i32< / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "n" > _< / span > < span class = "p" > ]< / span > < span class = "kt" > i32< / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "p" > }< / span > < span class = "w" > < / span > < span class = "p" > }< / span > < span class = "w" > < / span > < span class = "o" > **< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-33-8" name = "__codelineno-33-8" href = "#__codelineno-33-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-33-9" name = "__codelineno-33-9" href = "#__codelineno-33-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-33-10" name = "__codelineno-33-10" href = "#__codelineno-33-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-33-11" name = "__codelineno-33-11" href = "#__codelineno-33-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-33-12" name = "__codelineno-33-12" href = "#__codelineno-33-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-33-13" name = "__codelineno-33-13" href = "#__codelineno-33-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-33-14" name = "__codelineno-33-14" href = "#__codelineno-33-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ..< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "n" > i< / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-33-15" name = "__codelineno-33-15" href = "#__codelineno-33-15" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-33-16" name = "__codelineno-33-16" href = "#__codelineno-33-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-33-17" name = "__codelineno-33-17" href = "#__codelineno-33-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-33-18" name = "__codelineno-33-18" href = "#__codelineno-33-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-33-19" name = "__codelineno-33-19" href = "#__codelineno-33-19" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-34-1" name = "__codelineno-34-1" href = "#__codelineno-34-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.rs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-35-1" name = "__codelineno-35-1" href = "#__codelineno-35-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-35-2" name = "__codelineno-35-2" href = "#__codelineno-35-2" > < / a > < span class = "k" > fn< / span > < span class = "nf" > climbing_stairs_constraint_dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > : < span class = "kt" > usize< / span > < span class = "p" > )< / span > < span class = "w" > < / span > -> < span class = "kt" > i32< / span > < span class = "p" > {< / span >
< a id = "__codelineno-35-3" name = "__codelineno-35-3" href = "#__codelineno-35-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "k" > as< / span > < span class = "w" > < / span > < span class = "kt" > i32< / span > < span class = "w" > < / span > < span class = "p" > };< / span >
< a id = "__codelineno-35-4" name = "__codelineno-35-4" href = "#__codelineno-35-4" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
< a id = "__codelineno-35-5" name = "__codelineno-35-5" href = "#__codelineno-35-5" > < / a > < span class = "w" > < / span > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "k" > mut< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "fm" > vec!< / span > < span class = "p" > [< / span > < span class = "fm" > vec!< / span > < span class = "p" > [< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ];< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-35-6" name = "__codelineno-35-6" href = "#__codelineno-35-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-35-7" name = "__codelineno-35-7" href = "#__codelineno-35-7" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-35-8" name = "__codelineno-35-8" href = "#__codelineno-35-8" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-35-9" name = "__codelineno-35-9" href = "#__codelineno-35-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-35-10" name = "__codelineno-35-10" href = "#__codelineno-35-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-35-11" name = "__codelineno-35-11" href = "#__codelineno-35-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-35-12" name = "__codelineno-35-12" href = "#__codelineno-35-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "k" > in< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "o" > ..=< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-35-13" name = "__codelineno-35-13" href = "#__codelineno-35-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-35-14" name = "__codelineno-35-14" href = "#__codelineno-35-14" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-35-15" name = "__codelineno-35-15" href = "#__codelineno-35-15" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-35-16" name = "__codelineno-35-16" href = "#__codelineno-35-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-35-17" name = "__codelineno-35-17" href = "#__codelineno-35-17" > < / a > < span class = "p" > }< / span >
2023-07-01 22:39:28 +08:00
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > 在上面的案例中，由于仅需多考虑前面一个状态，我们仍然可以通过扩展状态定义，使得问题恢复无后效性。然而，许多问题具有非常严重的“有后效性”，例如：< / p >
< div class = "admonition question" >
< p class = "admonition-title" > 爬楼梯与障碍生成< / p >
< p > 给定一个共有 < span class = "arithmatex" > \(n\)< / span > 阶的楼梯，你每步可以上 < span class = "arithmatex" > \(1\)< / span > 阶或者 < span class = "arithmatex" > \(2\)< / span > 阶。< strong > 规定当爬到第 < span class = "arithmatex" > \(i\)< / span > 阶时，系统自动会给第 < span class = "arithmatex" > \(2i\)< / span > 阶上放上障碍物，之后所有轮都不允许跳到第 < span class = "arithmatex" > \(2i\)< / span > 阶上< / strong > 。例如，前两轮分别跳到了第 < span class = "arithmatex" > \(2, 3\)< / span > 阶上，则之后就不能跳到第 < span class = "arithmatex" > \(4, 6\)< / span > 阶上。请问有多少种方案可以爬到楼顶。< / p >
< / div >
2023-07-26 03:16:04 +08:00
< p > 在这个问题中，下次跳跃依赖于过去所有的状态，因为每一次跳跃都会在更高的阶梯上设置障碍，并影响未来的跳跃。对于这类问题，动态规划往往难以解决。< / p >
< p > 实际上，许多复杂的组合优化问题（例如旅行商问题）都不满足无后效性。对于这类问题，我们通常会选择使用其他方法，例如启发式搜索、遗传算法、强化学习等，从而在有限时间内得到可用的局部最优解。< / p >
2023-07-01 22:39:28 +08:00
< h2 id = "__comments" > 评论< / h2 >
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